Groups in which the centralizer of any non-central primary element is maximal
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Changguo Shao
Abstract
We investigate the structure of finite group G in which the centralizer of each non-central primary element of G is maximal in G. This provides an answer to the question raised in [X. Zhao, R. Chen and X. Guo, Groups in which the centralizer of any non-central element is maximal, J. Group Theory 23 2020, 5, 871–878], and also is a generalization of the result in [A. Mann, Finite groups with maximal normalizers, Illinois J. Math. 12 1968, 67–75]. In particular, we give an independent criterion of simplicity of a group, asserting that a group G is simple if the centralizer of each non-central primary element is maximal in G.
Funding source: National Natural Science Foundation of China
Award Identifier / Grant number: 12071181
Funding source: Natural Science Foundation of Shandong Province
Award Identifier / Grant number: ZR2020MA003
Funding statement: The authors are supported by the National Natural Science Foundation of China (No. 12071181) and the Nature Science Fund of Shandong Province (No. ZR2020MA003).
Acknowledgements
The authors would like to thank the referee for his/her valuable suggestions. It should be said that we could not have polished the final version of this paper well without his/her outstanding effort. The authors also would like to thank Professor A. Mann and N. V. Maslova for their valuable suggestions.
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Articles in the same Issue
- Frontmatter
- On the asymptotics of the shifted sums of Hecke eigenvalue squares
- On the splitting conjecture in the hybrid model for the Riemann zeta function
- Almost Dedekind domains without radical factorization
- Groups in which the centralizer of any non-central primary element is maximal
- GKM actions on cohomogeneity one manifolds
- Gruenberg–Kegel graphs: Cut groups, rational groups and the prime graph question
- Sobolev regularity for nonlinear Poisson equations with Neumann boundary conditions on Riemannian manifolds
- Higher differentiability for bounded solutions to a class of obstacle problems with (p,q)-growth
- Measures of noncompactness of interpolated polynomials
- The homotopy category of acyclic complexes of pure-projective modules
- The Gelfand–Kirillov dimension of Hecke–Kiselman algebras
- Sharp Li–Yau inequalities for Dunkl harmonic oscillators
- A Shimura–Shintani correspondence for rigid analytic cocycles of higher weight
- Free polynilpotent groups and the Magnus property
Articles in the same Issue
- Frontmatter
- On the asymptotics of the shifted sums of Hecke eigenvalue squares
- On the splitting conjecture in the hybrid model for the Riemann zeta function
- Almost Dedekind domains without radical factorization
- Groups in which the centralizer of any non-central primary element is maximal
- GKM actions on cohomogeneity one manifolds
- Gruenberg–Kegel graphs: Cut groups, rational groups and the prime graph question
- Sobolev regularity for nonlinear Poisson equations with Neumann boundary conditions on Riemannian manifolds
- Higher differentiability for bounded solutions to a class of obstacle problems with (p,q)-growth
- Measures of noncompactness of interpolated polynomials
- The homotopy category of acyclic complexes of pure-projective modules
- The Gelfand–Kirillov dimension of Hecke–Kiselman algebras
- Sharp Li–Yau inequalities for Dunkl harmonic oscillators
- A Shimura–Shintani correspondence for rigid analytic cocycles of higher weight
- Free polynilpotent groups and the Magnus property
